$2\mathrm{.}2.$ Vertical Differentiation with costly quality. Two firms are considering

entering a market. They will produce a special piece of equipment that is used

for metal fabrication, and both have the same capabilities, but their quality may

differ. Buyer $in[10,20]$ is willing to pay $s$ for a product of quality $s.$ The firms

can produce quality $sin[10,20]$ and have linear cost with constant marginal cost $c$

$($which you may assume to be $1,$ but should treat as variable for the best possible

grade$).$ As usual, they will make decisions over two periods. In the first period

they will simultaneously choose what level of quality to produce $($if at all$),$ and in

the second they will simultaneously choose price. Buyers are uniformly distributed

along $10,20.$

$(1)$ Write out your strategy for solving this problem.

$(2)$ Find a subgame perfect Nash equilibrium. Calculate profits.

$(3)$ Suppose that producing higher quality requires more investment. In partic$-$

ular, suppose that to be able to produce a product of quality $s,$ a firm must

invest $\frac{25}{8}{s}^2$ in period $1.$ What is the outcome now?